This grant is to begin a detailed study of the mechanisms underlying Ca2+ wave propagation. It is in two parts: (1) the study of spiral waves in Xenopus laevis oocytes; (2) the study of intercellular and intracellular waves in epithelial and glial cells. The first part will involve the detailed investigation and testing of a proposed model for wave propagation by Ca2+-induced Ca2+ release (Goldbeter et al., 1990). Previous work in the theory of biological excitability has focused on systems that behave in a similar fashion to the FitzHugh-Nagumo model for the propagation of an action potential. However, Goldbeter's model is an unusual excitable system and shows that realistic models of biological excitable systems do not always behave in this generic manner. The mathematical theory of these excitable systems is developed in order to derive new and elegant tests of the model. This theory results in a powerful test of the model that depends on the geometrical properties of the propagating spiral waves. A modification to the model, incorporating recent experimental evidence that [Ca2+] regulates IP3-induced Ca2+ release, is constructed, and the previous theory used to derive a test that will distinguish between the two versions of the model. The second part is the testing of a model for intercellular Ca2+ wave propagation in epithelial and glial cells. Experiments have shown that the Ca2+ wave is not mediated by local increases in Ca2+ concentration; it has been proposed that the wave is mediated instead by the diffusion of IP3. However, it is not clear whether the intercellular diffusion of IP3 can quantitatively account for the propagating wave. A quantitative realization of the model is constructed and solved in order to answer this question. The network will result in a reevaluation of current theories of biological excitability and the mechanisms of Ca2+ wave propagation.